Filling families and strong pure infiniteness

Abstract

We introduce filling families with matrix diagonalization as a refinement of the work by Rørdam and the first named author. As an application we improve a result on local pure infiniteness and show that the minimal tensor product of a strongly purely infinite $C^{\ast }$-algebra and a exact $C^{\ast }$-algebra is again strongly purely infinite. Our results also yield a sufficient criterion for the strong pure infiniteness of crossed products $A{⋊}_{\phi }\mathbb{N}$ by an endomorphism $\phi$ of $A$ (cf. Theorem 7.6). Our work confirms that the special class of nuclear Cuntz-Pimsner algebras constructed by Harnisch and the first named author consist of strongly purely infinite $C^{\ast }$-algebras, and thus absorb ${\mathcal{O}}_{\infty }$ tensorially.

39 pages. Notational text overlap with arXiv:1312.5195

Cited by in corpus (2)

• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• Crossed products by endomorphisms of $C_0(X)$-algebras